Lin Wng ,Feiying M ,Yongming He ,Dwei Liu

a Petroleum Engineering College,Guangdong University of Petrochemical Technology,Maoming,525000,China

b State Key Laboratory of Oil and Gas Reservoir Geology and Exploitation (Chengdu University of Technology),Chengdu,610059,China

Keywords:Non-homogeneous wettability Composite capillary walls Spontaneous imbibition Oil and water Lucas-Washburn equation

ABSTRACT In view of the classical Lucas-Washburn equation,which can only describe the spontaneous imbibition of single wetted capillary,a tilted composite capillary model with circular cross section,composed of different wettability capillary wall was established.The model can describe the spontaneous oil-water imbibition of water-wet capillary,oil-wet capillary and mixed wetting capillary.Through numerical solution of the model equation,it is found that the component content of the capillary walls,the capillary radius and the oil-water viscosity ratio have great effects on the spontaneous oil-water imbibition.Effects of capillary inclination angle and inertia force on spontaneous oil-water imbibition are related to the capillary scale.Effects of capillary inclination angle and inertia force can be ignored in small radius capillary,while effects of inclination angle and inertia force can not be ignored in large radius capillary.

1.Introduction

Spontaneous imbibition refers to the process of spontaneous displacement of non-wetting phase by the wetting phase under the action of capillary force.Natural porous media,such as rock,clay,etc.,consist of a matrix and pores that are often referred to as bundles of capillaries[1].The study of spontaneous imbibition can be used to evaluate the wettability and spontaneous imbibition capacity of solid and fluid systems[2,3].Spontaneous imbibition of capillary is widely used in groundwater engineering,petroleum engineering,textile dyeing,medical treatment and construction materials [4,5].

The dynamics of the spontaneous rise are described by many scholars.Lucas[6]and Washburn[7],ignoring effects of gravity and inertia forces,hypothesized that the pores of porous media are wethomogenized circular capillary bundles,and then first derived the analytic solution for the rise of water in capillaries by combining Hagen-Poiseille equation and Young-Laplace's equation.Lucas-Washburn equation writes:

where t is the time,h（t）is the rising height of the meniscus at time t,σ is the surface tension,θ is the contact angle,μwis the viscosity of water,R is the capillary radius.

The derivative of Equation (1) with respect to time can be obtained as follows:

As can be seen from Equation (2),the velocity obtained in Equation (2) is infinite when time is zero,and the actual initial velocity should be zero when time is zero,so the Lucas-Washburn equation cannot accurately describe the actual physical situation.However,the Lucas-Washburn equation has been widely used in the spontaneous imbibition of porous media due to its simplicity.The Lucas-Washburn equation is based on some ideal assumptions,which limits the applicability of the equation.Later scholars made some improvements based on the Lucas-Washburn equation.

In 1971,the Lucas-Washburn equation was derived by Szekely et al.[8]in terms of energy balance,solving the problem that the previous Lucas-Washburn equation could not be calculated within a short period of time at the beginning of water rise (when time approaches 0,the velocity becomes infinite).It writes

where ρ is density,g is acceleration of gravity.

In 2000,Zhmud et al.[9]took the gravity into account and further developed the Lucas-Washburn equation to obtain the asymptotic solution of the rising height of the liquid-gas interface changing with time.

The asymptotic solution corresponding to the short-time limit(t→0) is given by

The asymptotic solution corresponding to the long-time limit is given by

where h∞is the height of the rising meniscus when it is about to approach stable level.

In 2002,Benavente et al.[4]improved Lucas-Washburn equation by taking the shape factor of pores and the tortuosity of capillaries into account,and obtained the following equation:

where W（t）is the weight of the water,A is the cross-sectional area of a porous medium,φ is the porosity,Reffis the effective radius of capillary,δ is the shape factor of capillary section,ζ is the tortuosity.

In 2008,Fries and Dreyer [10]studied the equation of capillary fluid rise taking the gravity into account.They introduced Lambert W function to further develop the Lucas Washburn equation,and the analytic solution (Equation (7)) obtained was more applicable to the calculation of capillary fluid rise over a longer period than the classical Lucas-Washburn.However,the method ignores the effect of inertial forces.It writes:

In 2015,Berthier et al.[11]studied the spontaneous imbibition of composite capillary for any cross section and concluded that the Reynolds number was so small that the inertial force could be ignored.Ignoring the effect of gravity,they obtained a relation between the distance of the interface in the capillary and time,which writes:

where f is the Fanning coefficient,Re is Reynolds number,DHis the hydraulic radius.

Capillaries,both in the original and modified Lucas-Washburn equation,were regarded as homogeneous wetting.However,in porous media such as underground rock,the wettability of pores is non-homogeneous.Real rocks are composed of different minerals.There are differences in their wettability degrees,such as strong,weak,neutral and even oil-wet.Therefore,rocks are nonhomogeneous and wettability varies with spatial location [12,13].For example,in shale reservoirs,there are both hydrophilic minerals(quartz,feldspar,calcite,etc.)and oil-wet substances(organic matter,pyrite,etc.)[14].These pore walls are composed of minerals with different wettability.When immiscible two-phase fluid moves in such pores,the imbibition dynamics of non-homogeneous pores are different from those of homogeneous wetting pores.In addition,the capillaries in the underground rocks may be tilted in some extent not horizontal or vertical.Based on the classical Lucas-Washburn equation,a straight composite capillary model with circular cross section,composed of different wettability substances,was established to quantitatively describe the oil-water spontaneous imbibition.

2.Spontaneous oil-water imbibition equation of the tilted composite capillary model

2.1.Model description

The pores in the rock are simplified into straight capillaries with a circular cross-section of equal diameter.The walls of the capillaries are composed of different material components,as shown in Fig.1.The capillary is placed at a certain inclination angle in the absence of external force on the fluid.

2.2.Spontaneous oil-water imbibition equation of composite capillary

Before establishing the equation,some assumptions should be made as follows:

(1) The fluid is laminar;

(2) The fluid is Newtonian fluid;

Fig.1.Schematic diagram of composite capillary.

(3) The compressibility of the fluid is negligible;

(4) Minerals are arranged in parallel on the capillary wall;

(5) The capillary is saturated with oil in the initial state.

The fluid in the capillary is affected by surface tension Ft,viscous force Ff,gravity Fg,as shown Fig.2.Ftigenerated by interaction between capillary wall i and oil-water:

where Ftiis the capillary force generated by the wall component i and oil-water system,fiis areal fraction of the wall component i in total wall,θiis the Young's contact angle with each component i,σwois oil-water interfacial tension.

The resultant force (capillary force) in the X-axis direction caused by interfacial tension is:

where Ftis capillary force,n is the total number of components.

The average contact angle of the capillary walls with n materials is [15-17]:

θ is the average contact angle.Equation (11) is actually the expression for the generalized Cassie contact angle [18].

By substituting Equation (11) into Equation (10),we can get:

The viscous force of fluid subjected to the inner wall of the capillary is:

where Ffis viscous force,x is the motion distance of the interface,τwis the shear stress of water and capillary wall,τois the shear stress of oil and capillary wall,L is the length of the capillary.

Fig.2.Fluid force diagram in capillary.

When the fluid is laminar in a circular cross section capillary,the viscous shear stress can be written as

where τ is shear stress between fluid and capillary wall,μ is viscosity,v is the average velocity on the cross section of the capillary.Substituting Equation (14) into Equation (13),we can get：

The component force of gravity in the x-axis direction is:

where Fgis the gravity,m is the mass of the liquid,α is the inclination angle of the capillary.

The mass of the fluid in the capillary is:

where ρwis the density of water,ρois the density of oil.

Substituting equation (17) into equation (16),we can get:

It can be obtained by Newton's second law:

where a is the acceleration.

By substituting equations(12),(15)and(18)into equation(19),we can get:

By substituting equation (21) and equation (22) into equation(20),we can get:

In equation (23),the first term represents inertia force,the second viscous force,the third gravity,and the fourth capillary force.Equation (23) could just get the numerical solution not the analytic one.

When the average contact angle is less than 90°,the initial condition of Equation(23)is:when t=0,x=0 and v=0.Therefore,the initial condition can be written as:

When the average contact angle is greater than 90°,the initial condition of Equation (23) is:

3.Numerical simulation verification

3.1.Level set method

The level set method can be used to track moving interface of two immiscible phases[19].The level set function(∅)is a smooth step function.The region where ∅=0 is filled with oil and the region where ∅=1 is filled with water.The fluid interface is represented by the 0.5 contour of the function ∅.The transport of the fluid interface is given by

where ∅is level set function,γ is intensity of reinitialization,ε is control parameters of interface thickness.

The density and viscosity can be respectively expressed as follows [20]:

where ρ is the density of fluid,μ is the dynamic viscosity of fluid,ρois the oil density,ρwis the water density,μoand μware the dynamic viscosity of oil and water respectively.

3.2.Governing equations

The governing equations consist of the continuity equation and the incompressible Navier-Stokes equation coupled with the level set equation [21,22].

Continuity equation:

Taking account of capillary effects,the Navier-Stokes equation is then:

where p is the pressure,I is the identity matrix,Fstis the surface tension force.

The surface tension force can be expressed as [20,21]:

where k is curvature,δ is Dirac delta function,n is the unit normal to the interface.

The delta function δ can be approximated as [21,22]:

Interface normal n,curvature k can be respectively expressed as follows [20-22]：

3.3.Numerical simulation verification

To validate the present model,the numerical simulation is performed in the two-phase flow module of COMSOL Multiphysics soft by using the level set method.The geometric model is a cylindrical composite capillary,as shown in Fig.3.The capillary walls are composed of organic matter(f1=25%),quartz(f2=25%),calcite(f3=25%).Contact angles of capillary walls are respectively 120°,20°,30°.The capillary has a radius of 100 μm and a length of 1000 μm.Inclination angle of the composite capillary is 60°.The viscosity of water is 1 cP,and oil 5 cP.The density of water is 1000 kg/m3,and oil 800 kg/m3.The oil-water interfacial tension coefficient is 0.05 N/m.The pressure of capillary inlet and outlet are 0.In the simulation the parameter γ defining the intensity of reinitialization is 0.20 m/s,and the parameter ε controlling interface thickness is 9 × 10-6m.

The spontaneous imbibition velocity of numerical simulation has been compared with results of the present model (Fig.4).As shown in Fig.4,the discrepancy between the two curves (the present model and numerical simulation) is very small.

4.Results and discussions

The model takes effects of capillary force,viscous force,inertia force and gravity into account.It is suitable for the spontaneous oil-water imbibition of hydrophilic,oil-wet and mixed wetting pores.The model has good adaptability to spontaneous imbibition description of shale and other reservoir rocks with mixed wetting pores.In order to analyze the spontaneous oil-water imbibition in the model,the data in Table 1 are used for calculation.

Fig.3.Composite capillary geometry.

4.1.Effect of composition content of capillary walls on spontaneous oil-water imbibition

Fig.4.Comparison between numerical simulation and the present model for spontaneous imbibition velocity.

The average contact angle of the capillary model can be obtained from equation (11):

According to equation (33),the average contact angle of composite capillaries is related to the component content of capillary walls and their contact angles.The wettability of reservoir rocks is also heterogeneous due to the different component contents of minerals at different locations.Equation (33) is used to obtain the average contact angle of composite capillary walls in Table 2.Then,according to equation (23),curves of spontaneous oil-water imbibition under different mineral component contents are drawn(see Fig.5).When the wettability of the capillary is water-wet,the dominant force of spontaneous imbibition is capillary force,and the resistances are gravity and viscous resistance.When the wettability of capillary is oil-wet,the dominant forces of spontaneous imbibition are capillary force and gravity,and the resistance is viscous force.As can be seen from Fig.5,the higher the oil-wet component content of capillary walls is,the lower the capillary spontaneous imbibition rate is.When the content of oil-wet component contents of capillary walls is high,the average wettability of the capillary is oil-wet,which is not conducive to the production of water drive oil reservoir.

4.2.Effect of oil-water viscosity ratio on spontaneous imbibition

Oil-water viscosity ratio can effect spontaneous imbibition dynamics[23].After changing the viscosity of the oil,the parameters in Table 1 were substituted into equation (23),and curves of spontaneous imbibition under different oil-water viscosity ratios could be obtained,as shown in Fig.6.

As can be seen from Fig.6,when the viscosity ratio of the oil to the water is equal to 1,the velocity of spontaneous imbibition remains constant.When the oil-water viscosity ratio is less than 1,the velocity of spontaneous imbibition decreases gradually.When the oil-water viscosity ratio is greater than 1,the velocity of spontaneous imbibition increases gradually.The velocities of spontaneous imbibition at the end point of capillary are the same for all curves under different oil-water viscosity ratios.

4.3.Effects of scale on spontaneous oil-water imbibition

4.3.1.The effect of capillary radius on spontaneous oil-water imbibition

After changing capillary radius,the parameters in Table 1 were substituted into equation (23),and curves of spontaneous imbibition under different capillary radius could be obtained,as shown in Fig.7.In Fig.7,the change trend of the spontaneous imbibition rate versus x is different.The curve with a radius of 1000 μm is convex,while the other (1 μm,10 μm,100 μm) is concave.It is because spontaneous imbibition rate is determined by both driving force(capillary force)and resistance(viscous resistance and gravity).The proportion of driving force and resistance varies at different stages.

For capillaries with radius of 1 μm,10 μm,100 μm,their fluid mass is very little due to their very small radius,and their gravities are negligible compared to capillary forces.With their spontaneous imbibition process,the volume of water phase increases and the volume of oil phase decreases.It leads to viscous resistance decrease,and the proportion of resistance decreases.This eventually causes the acceleration of fluid movement to increase,and the curves become convex.

For capillary with a radius of 1000 μm,the fluid mass is obviously larger than that of the other three capillaries due to the larger radius,and the gravity effect can not be ignored.With the fluid imbibition process,the volume of water phase increases and the volume of oil phase decreases.It leads to the viscous resistance decrease,but the gravity increases,and the total resistance increases.This causes the acceleration of fluid movement to decrease,and the curve becomes concave.

4.3.2.Effect of capillary inclination angle on spontaneous oil-water imbibition

After changing capillary radius and inclination angle,the parameters in Table 1 were substituted into equation (23).Under capillary radius of 100 μm and 1000 μm,curves of spontaneous oilwater imbibition at different capillary inclination angle could be obtained,as shown in Fig.8.As can be seen from Fig.8 in these curves under capillary radius of 100 μm,the velocity remains almost unchanged when capillary inclination angle changes.In these curves under capillary radius of 1000 μm,the velocity varies significantly with increasing capillary inclination angle,which showed that the angle of inclination positively effects the velocity.This indicates that the sensitivity of inclination angle to the velocity is related to the capillary scale,and the larger the capillary radius is,the greater the effect of inclination angle is.The main reason for this phenomenon is that the larger the capillary radius is,the greater the component of gravity in the axial direction of the capillary will become.The capillary inclination angle affects thecomponent of gravity in the axial direction of the capillary,thus the inclination angle affects the velocity of spontaneous oil-water imbibition.Kewen Li [24]analyzed the application criterion of Lucas-Washburn equation and concluded that when the ratio of capillary force to gravity was greater than 3,the effect of gravity on the velocity of spontaneous imbibition could be ignored.For capillaries below micron scale,the effect of gravity is much smaller than capillary force and viscous resistance,and the effect of capillary inclination angle can be ignored.

Table 1 Capillary and fluid parameters.

Table 2 Component contents of capillary walls.

Fig.5.Curves of spontaneous oil-water imbibition under different component contents.

Fig.6.Curves of spontaneous imbibition at different viscosity ratios.

Fig.7.Curves of spontaneous imbibition at different capillary radius.

4.3.3.Effect of inertia force on spontaneous oil-water imbibition

Because the inertial force belongs to the virtual mass force,and the fluid mass in the capillary is related to the capillary radius,so the capillary radius affects the inertial force.Curves of spontaneous oil-water imbibition with and without inertia are drawn in Fig.9.As can be seen from Fig.9,the larger the capillary radius,the greater the effect of the inertial forces.When the capillary diameter is less than 50 μm,the effect of inertia force is small.When the diameter is greater than 50 μm,the inertia force will have a significant effect.

In addition,it can be found from Fig.9 that at the early stage of spontaneous imbibition,the difference between considering the inertial force and not considering the inertial force is relatively small,but at the later stage the difference increases.This is because the height of the meniscus is small at the early stage of spontaneous imbibition.As spontaneous imbibition goes on,the height becomes larger and larger.This will lead to an increase in fluid mass,so does the inertial force at the later stage.Therefore,the spontaneous imbibition of capillary with a small radius can be approximately considered to be in a state of force equilibrium,and the spontaneous imbibition of capillary with a small radius can be described using the classic Lucas-Washburn equation.The spontaneous imbibition of capillary with large radius is affected by the inertia force,so the spontaneous imbibition of capillary should be described under the non-equilibrium state of force.

5.Conclusions

(1) The component content of the capillary walls affects the average contact angle and dynamics of spontaneous oilwater imbibition.

(2) The oil-water viscosity ratio has a great effect on spontaneous oil-water imbibition.When the viscosity ratio is equal to 1,the velocity of spontaneous oil-water imbibition basically keeps constant.When the viscosity ratio is less than 1,the velocity of spontaneous oil-water imbibition decreases gradually.When the viscosity ratio is greater than 1,the velocity of spontaneous oil-water imbibition increases gradually.

(3) Effects of capillary inclination angle and inertia force on velocity of spontaneous oil-water imbibition are related to capillary scale.The larger the capillary radius is,the more obvious effect of capillary inclination angle and inertia force on spontaneous oil-water imbibition will be.Therefore,effects of capillary inclination angle and inertia can be ignored in small radius capillary,while effects of inclination angle and inertia force should be considered in large capillary.

Fig.8.Curves of spontaneous imbibition at different capillary inclination angles.

Fig.9.Curves with and without inertia force at different scales.

Acknowledgments

Support from Open Fund(PLC20190804)of State Key Laboratory of Oil and Gas Reservoir Geology and Exploitation (Chengdu University of Technology),Maoming City Science and Technology Plan Project (2019413 and 2020530),Guangdong Natural Science Fund(2019A1515012235),China National Fund for Studying Abroad(201808440643) are gratefully acknowledged.

Appendix A.Supplementary data

Supplementary data to this article can be found online at https://doi.org/10.1016/j.petlm.2021.07.001.

Author statements

Under guidance of Yongming He,Lin Wang established the model,derived the equation and performed calculations.Feiying Ma did the drawing and data analysis.Dawei Liu modified the language.All authors read and contributed to the manuscript.